6970
Ind. Eng. Chem. Res. 2003, 42, 6970-6976
Modeling Alkaline Silicate Solutions at 25 °C Charles F. Weber* and Rodney D. Hunt Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, Tennessee 37831-6370
A thermodynamic model is developed to describe silicate behavior in solutions containing high concentrations of both NaOH and NaNO3. Experimental solubility data in such solutions are also presented. The model itself applies Pitzer’s ion-interaction approach and involves several aqueous polymeric species. Model predictions compare well with a variety of experimental datas solubilities, NMR results, and pH measurements. Introduction
molecule: Additional polymerization occurs through
As a result of both defense and commercial activities, the U.S. Department of Energy has accumulated a large volume of inorganic radioactive waste. The presence of silica species in tanks or waste streams poses significant problems in waste processing for both Hanford and Savannah River sites. For example, the condensed offgas stream from the vitrifier at Savannah River is rich in NaNO3 and silicon and is slightly acidic. To protect mild steel piping and tanks, this waste stream must be neutralized with NaOH prior to storage and subsequent evaporation. By itself, silica is quite soluble in alkaline solutions; however, in the presence of aluminum or other waste components, its solubility decreases markedly. For example, Savannah River has experienced numerous problems with precipitation of alumino-silicates in evaporators and related pipes and tanks. An accurate model of silicate behavior in complex solutions is essential in designing safe and effective processes for waste handling. Because virtually all the wastes contain high concentrations of nitrate and hydroxide, we consider here the modeling of silicate solubility in solutions containing both these ions.
analogous bonding of multiple OH groups which form chains, rings, or 3-dimensional structures. Many of these shapes have been identified through NMR spectroscopy and other techniques. The hydroxyl groups ionize to form silicate anions as illustrated below for the monomer:
Silicate Hydrolysis It is well-known that the solubilities of both quartz and amorphous silica increase as the solution alkalinity increase above a pH of 8.1 This solubility effect is true in pure hydroxide solutions2 and in the presence of other salts.3 It is also understood that considerable polymerization of the silicate species occurs in the pH range of 10.8 to 12.2. Above a pH of 12.2, polymerization declines as solubility continues to increase. In aqueous solutions, solid silica hydrolyzes to form monomeric silicic acid:
SiO2 + 2H2O T Si(OH)4.
(1)
Dimerization joins two silicon atoms through a common bond with an oxygen atom and releases a water * To whom correspondence should be addressed. Tel.: (865) 576-4475. Fax: (865) 576-3513. E-mail:
[email protected].
Si(OH)4 T H+ + SiO(OH)3-
(3)
SiO(OH)3- T H+ + SiO2(OH)22-
(4)
As alkalinity increases, ionization also increases; in high pH solutions, concentrations of silicate anions greatly exceed that of hydroxide itself. To simplify terminology, a shorthand notation is used to represent the polymeric species
(i, j) ) SijOm(OH)ni-
(5)
where m and n vary according to the bonding arrangements of the various species. Several common examples involving regular and symmetric species are shown in Table 1. It is also useful to define the negative charge per Si atom for a given molecule:
Z ) i/j For the solutions as a whole, Z h denotes the total accumulated charges from all silicate ions divided by the total number of Si atoms in solution. Each of the examples in Table 1 is symmetric; however, this need not be the case. The processes of dissolution, polymerization, and ionization create a tremendously complicated system in which hundreds of individual species may exist. Modeling such a system requires simplification, empiricism, and even heuristic reasoning. Furthermore, it is desired to integrate silicate
10.1021/ie0303449 CCC: $25.00 © 2003 American Chemical Society Published on Web 11/14/2003
Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 6971 Table 1. Common Silicate Polymers
Table 2. Experimental Values on the Amorphous Silica Solubilities reference
type of data
pH range
data points
max I (m)a
2 3
NaOH NaOH + 1 M NaCl NaOH + 3 M NaCl NaNO3 NaOH + 1 M NaNO3 NaOH + 3 M NaNO3
8-10.7 9-10.7 8.8-10.3 ∼7 6-11.5 6-11.5
12 18 14 9 10 10
0.1 1 3 6.12 2 4
7 this work
a Considering only NaOH and NaNO and assuming complete 3 dissociation.
ions. These effects were measured for many common salt solutions,7,8 and some of the results are listed in Table 2. Alexander et al.2 have measured silicate solubility in caustic solutions at 25 °C; similar results of many researchers are summarized by Eikenberg.9 Zarubin and Nemkina3 give solubilities at 25 °C in 1 M and 3 M NaCl solutions as a function of pH. In this work, we present solubility results for solutions of both NaOH and NaNO3, as described in the next section. Experimental Approach n ) 0, 1, 2, ... b Except for the dimer, all oxygen linkages are denoted by lines. If not shown, silicon atoms occur at vertices. OH and O- groups are not shown except for the dimer. a
behavior with other electrolyte components, which are to be modeled using Pitzer’s ion interaction treatment. As seen from the examples in Table 1, each silicon atom is connected via an oxygen bridge to as many as four other silicon atoms. Each silicon atom is classified according to its “connectivity type”, the number of other silicon atoms to which it is connected in the molecule. NMR measurements generally determine the presence of these different connectivity types, labeled Q0, Q1, Q2, Q3, and Q4. For example, Q0 represents the total inventory of species in which no silicon atom is adjacent to another (monomeric species). The dimer and end groups of other linear molecules comprise the Q1 group, since each is connected to exactly one other silicon atom. The group Q4 represents large polymers in which virtually every silicon atom connects to another; this could also represent precipitated amorphous silica or colloidal solids. Interior atoms of linear molecules and the cyclic species correspond to Q2, where each silicon atom is connected to two others. However, the sharp bond angles in the cyclic trimer produce a separate NMR signal, which is denoted as Q∆2. Analogously, the groups Q3 and Q∆3 correspond to connectivities with three other silicon atoms, such as those in the prismatic octamer and hexamer, respectively. Sometimes it is possible for the NMR signal to distinguish other minor differences within connectivity groups. However, such determinations are highly uncertain and contribute little to model capabilities. Therefore, this study was only concerned with the following basic connectivity groups: Q0, Q1, Q2, Q∆2, Q3, Q∆3, and Q4 as represented by the regular structures in Table 1. Data are appropriated from Svensson et al.,4 McCormick et al.,5 and from Kinrade and Swaddle.6 Solubilities Silicic acid is sparingly soluble near neutral pH, and this behavior is affected slightly by the presence of other
Each solubility sample consisted of deionized water, sodium nitrate (Aldrich Chemical Co.), sodium hydroxide (VWR), and silicon dioxide in the form of fumed silica (Alfa Aesar). Fumed silica was selected as the source of silicon due to its very small particle size, which should permit faster equilibration. All samples were prepared at 25 °C in a graduated polyethylene cylinder. After the required amount of sodium nitrate was weighed and transferred to the graduated cylinder, nearly all of the required water was also added. The sodium nitrate was then dissolved, and the pH of the sample was adjusted with sodium hydroxide (pH was monitored with an Orion 920A pH meter). After the fumed silica was added, a small amount of water was added to the sample to obtain the desired volume. The solutions were then transferred and sealed in high-density polyethylene bottles. The amounts of fumed silica in the tests were initially 20% more than crude estimates of solubility. If no silicon solids could be observed, then a small amount of silicon dioxide (an additional 10-20%) was added to the samples. This addition was repeated until solids were observed. With a few samples, the amount of fumed silica was reduced due to an excess amount of solids. After the final addition of fumed silica, the samples were sealed and heated to 35 °C for 1 week. Following this initial equilibration period, the temperature of the samples was reduced to 25 °C. The pH of the solution was used to monitor the progress toward equilibrium, which took approximately 6 weeks. Most samples were then filtered using two syringe filters, which were 0.8/0.2 Fm Supor7 membrane (Gelman Laboratory) and 0.2 Fm Anotop 25 inorganic membrane (Whatman). The samples with an initial pH of 14 could only be filtered with the 0.8/0.2 Fm Supor7 membranes. After the density of the filtered solution was determined, the sample was diluted in a polyethylene bottle. The soluble silicon concentration was measured through the use of inductively coupled emission spectroscopy (ICP). The ICP for this silicon analysis was a Model 61E Trace Analyzer (Thermo Jarrell Ash), which is a simultaneous plasma emission spectrometer. The silicon concentrations (listed in Table 3) were
6972 Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 Table 5. Pitzer Parameters at 25 °C
Table 3. Experimental Results initial OH(mol/L)
final pH
soluble Si (mol/L)
density (g/mL)
data Z
0.000011 0.000052 0.000104 0.000516 0.00103 0.00509 0.0101 0.100 0.494 0.981
1 M NaNO3 at 25 °C 6.17 0.00223 7.12 0.002 7.25 0.00192 8.13 0.00214 8.4 0.00206 9.34 0.00243 9.7 0.00469 10.94 0.0634 11.12 0.253 11.58 1.42
1.052 1.052 1.052 1.052 1.052 1.052 1.052 1.058 1.077 1.138
0.005 0.026 0.054 0.241 0.498 2.093 2.155 1.571 1.952 0.691
0.000011 0.000052 0.000104 0.000516 0.00103 0.00509 0.0101 0.100 0.494 0.981
3 M NaNO3 at 25 °C 5.97 0.0014 6.88 0.00124 7.05 0.00123 7.92 0.0015 8.16 0.00136 9.06 0.00165 9.46 0.00286 10.76 0.0267 11.1 0.19 11.61 1.08
1.157 1.157 1.157 1.157 1.157 1.157 1.157 1.162 1.179 1.233
0.008 0.042 0.085 0.344 0.754 3.083 3.533 3.729 2.600 0.909
Table 4. Silicate Species and Gibbs Energies of Formation species (0,1) (1,1) (2,1) (2,2) (4,2) (6,3) (2,4) (4,4) (6,6) (4,8) (8,8) SiO2 H2O H+ Na+ NO3OHa
formula Si(OH)4 SiO(OH)3SiO2(OH)22Si2O3(OH)42Si2O5(OH)24Si3O6(OH)33Si4O6(OH)62Si4O8(OH)44Si6O156Si8O16(OH)44Si8O208SiO2
name monomer monomer monomer dimer dimer cyclic trimer cyclic tetramer cyclic tetramer prismatic hexamer prismatic octamer prismatic octamer amorphous solid
NMR group Q0 Q0 Q0 Q1 Q1 Q∆2 Q2 Q2 Q∆3 Q3 Q3
species Na+
Na+ Na+ Na+ Na+ Na+ Na+ Na+ Na+ Na+ Na+ Na+ NO3NO3NO3NO3NO3NO3NO3NO3NO3NO3NO3NO3-
(1,1) (2,1) (2,2) (4,2) (6,3) (2,4) (4,4) (6,6) (4,8) (8,8) NO3OH(0,1) (1,1) (2,1) (2,2) (4,2) (6,3) (2,4) (4,4) (6,6) (4,8) (8,8) OH-
β(0)
β(1)
C
referencea
0.7924 -0.1145 -0.0893 -0.0204 0.0050 0.0268 -0.3131 -0.2922 -0.4775 -0.3233 0.00204 0.0869 0.05b 0c 0.4c 0.4c 0.4c 0.4c 1.2c 0.6c 0.6c 1c 0.8c -0.0547c
-4.202 -9.1375 -7.3034 -12.0415 -12.2557 -6.739 -3.4868 -0.2316 -2.3039 -8.4712 0.2368 0.2481
0.0488 -0.0102 -0.0273 -0.0185 -0.0192 -0.0418 -0.0161 -0.0159 -0.015 0.0453 0.00008 0.0039
18 17
a
22 b
Derived in this work if no reference given. This parameter describes interaction between ion and neutral species. c This parameter is θ (interaction between ions of like charge).
µ°/RT
referencea
-528.318 -504.633 -469.704 -451.519 -420.494 -372.594 -417.945 -406.861 -361.081 -372.149 -356.300 -343.128 -95.667 0 -105.642 -44.707 -63.446
12
replaced with a new electrode. In either case, the previous pH measurements were repeated. Third, the concentration of hydrogen ions is very small in solutions with a pH of 10 and above, and the measurement of very small concentrations can be inherently difficult. To determine the potential degree of error, a calibrated electrode was used to determine the pH of samples with known hydrogen ion concentrations. The test results indicated that the reported pH values should be within 0.1 pH units.
19 19 19 19
Derived in this work if no reference given.
determined through both direct analysis and standard silicon additions, which yielded comparable results. An Orion Model 920A pH meter with an Orion Ross 8103 pH electrode was used to determine the pH of the solutions. Prior to each use, the meter was always calibrated with two solutions of known pH. For the solutions with a pH greater than 10, a Thermo Orion Application buffer solution with a pH of 10.01 was used as one of the calibration solutions. The other solution was either a 0.1 N sodium hydroxide solution (J. T. Baker) or a 1 N sodium hydroxide solution (VWR). It is important to note that three potential problems can occur with the use of these sodium hydroxide solutions. First, the sodium hydroxide solution can react with carbon dioxide from the air to form carbonate and lower the hydroxide concentration. This concern was minimized through the use of several new sodium hydroxide bottles. Second, the sodium hydroxide will attack the glass membrane in the pH electrode over time. Periodically, the performance of the electrode was checked using one of the calibration solutions. If a significant difference in the pH readings for the same solutions was observed, the electrode was either recalibrated or
Modeling Silicate Species The most common approach to modeling silicate species in highly caustic solutions is to select several species as representative samples and to estimate empirical formation constants. One study assumed only the monomer and the dimer, but it allowed all possible ionizations (even those known not to occur).10 Most other modelers4,9,11-13 have included a few higher polymeric species as well, since such an approach is more comprehensive. Recognizing that dozens of polymeric species exist, some simplification is also justified. Hence, a single representative polymer for each connectivity group was selected. With the exception of the linear trimer, linear tetramer, and generic polymer, all the polymers in Table 1 were used in this study. In evaluating which anions must be included, it is helpful to observe the general trends in the data of Svensson et al.4 When data sets having similar amounts of dissolved silicon are grouped together, the changes in each of the NMR connectivity groups can be plotted against pH, as shown in Figure 1. Plots a, b, and c indicate that connectivity groups Q0, Q1, and Q∆2 all are increasing as pH increases. This suggests that the monomer, dimer, and cyclic trimer have highly charged species (they may have other species as well); hence, species with Z ) 2 are included for these shapes. In addition, the level curves for Q0 and Q1 in the pH range 11-12 suggest species of lower charge, too, so species with Z ) 1 must also be included. Plots d and e indicate
Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 6973
Figure 1. Comparison of NMR data at various pH values. (a) Q0, (b) Q1, (c) Q2, (d) Q∆2 (e) Q∆3 (f) Q3. Data from ref 4. Lines connect equilibrium points of similar silica concentrations.
that connectivity groups Q2 and Q∆3 rise initially, then drop, suggesting no highly charged species. We consider species with Z ) 0.5 and Z ) 1 for the cyclic tetramer and prismatic hexamer (in practice, only one species was required for the latter, with Z ) 1). Finally, for the connectivity group Q3, plot f indicates only decrease with increasing pH. Again we consider the lower charged species with Z ) 0.5 and Z ) 1 for the cubic octamer. Evidently, this species increases in abundance rapidly in the pH range from 10.7 to 11.4, below the range of the Svensson data. The model presented here calculates phase equilibrium by minimization of total Gibbs energy. The method is based on the code SOLGASMIX, which is described elsewhere in detail.14,15 Necessary input parameters are the Gibbs Energy of Formation for each species and activity coefficient parameters for aqueous species. The Pitzer procedure is used to calculate activity coefficients, requiring binary parameters β(0), β(1), and C for each cation-anion pair. In addition, accurate solubility predictions usually require mixture parameters θ (between ions of like charge) and/or ψ (involving three ions, not all with same charge). The details of Pitzer’s model can be found in refs 16 and 17.
A fairly extensive database has been developed over the course of many studies on waste processing. Some of the Gibbs Energies and Pitzer parameters have been taken from the literature or our own previously published work. Others are determined from regressions of data as a part of this work. All are given in Tables 4 and 5 and the references noted there. The Pitzer method involves the use of ionic strength. When dealing with ionized polymers, the exact definition of ionic strength becomes clouded. Throughout this work we assume a calculation of ionic strength based on the charge per silicon atom rather than on the total charge of each polymeric molecule. This is somewhat more realistic since the cubic octamer with a charge of -1 per silicon atom should not contribute to the ionic strength as if it were a single ion of charge -8. It could easily be argued that the charge contribution should count for more than -1 as well. An additional problem arises if the charge per silicon atom is not an integer. In this case, ionic strength can still be calculated, for example, using a charge of -1/2. These difficulties arise because the concept of ionic strength was not developed with polymeric ions in mind. In practice, the use of
6974 Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003
Figure 2. Q0 connectivity group (monomer). Data from refs 4-6. Units of both axes are mol/kg.
charge per Si atom produces better fits to data and is therefore justified for that reason. The parameters determined in the course of this work are Gibbs energies for aqueous silicate ions and Pitzer parameters for interactions between Na+ and silicate ions. The Gibbs energy for aqueous silicic acid (0,1) was taken from ref 12 since this work is a consensus standard. A tentative value for the Gibbs energy of amorphous SiO2 solid was determined using only solubility data between pH 7 and 8, and the (0,1) aqueous species. Gross estimates of other Gibbs energies were determined heuristically using solubility and NMR data. These estimates were refined in subsequent regressions. Initial estimates for Pitzer parameters were either chosen heuristically or simply assigned the value zero. Final regression included all parameters optimized simultaneously. A special version of the code has been developed which optimizes selection of parameter values through nonlinear regression (i.e., minimization of total squared error) of data. Relevant experimental data have included solubilities, activity or osmotic coefficients, NMR connectivity distributions, and pH values. During the parameter estimation process, several combinations of species were used. The resulting set of species is based on the fewest number which could adequately describe all experimental data. A few aqueous ions were eliminated, either because their concentrations were extremely small or their inclusion did not lower the sum of squared error appreciably. However it is important to note that a different set of representative ions might also match the data well. That is, the species used, and the parameters that describe their interactions, are not necessarily unique. However the derived parameters are empirically connectedsthere are several strong correlations between them. Nevertheless, the species and parameters used here do represent a credible model capable of depicting a wide variety of dataspH, NMR, and solubility.
Figure 3. Q1 connectivity group (dimer). Data from refs 4-6. Units of both axes are mol/kg.
Figure 4. Q∆2 connectivity group (cyclic trimer). Data from refs 4-6. Units of both axes are mol/kg.
Figure 5. Q2 connectivity group (cyclic tetramer). Data from refs 4-6. Units of both axes are mol/kg.
Model Predictions and Discussion Predictions of speciation in silicate solutions at 25 °C are compared with NMR data in Figures 2-7. In each figure, both axes represent concentrations of the various connectivity groups (mol/kg water); a perfect match between data and calculation would lie on the diagonal line. While there is some scatter, the model generally predicts each of the connectivity groups fairly well. We note that many of the solutions involved total silica and NaOH concentrations above 10 M, the highest being
Figure 6. Q∆3 connectivity group (prismatic hexamer). Data from refs 4-6. Units of both axes are mol/kg.
about 20 M in each. The model does an excellent job with the more dominant groups Q2 and Q3, even to 10 M in these species. The calculated values of pH are compared with measured values in Figure 8. Again there is some scatter, but the predictions probably do not exceed the uncertainty of the data. This includes not only the
Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003 6975
Figure 7. Q3 connectivity group (prismatic octamer). Data from refs 4-6. Units of both axes are mol/kg. Figure 10. Silicate solubility in NaOH solutions. - Calculation, [ Data from ref 2.
Figure 8. Comparison of calculated and measured pH. Data from refs 4 (]) and 5 (4).
Figure 11. Silicate solubility in solutions of 1 M NaNO3 and variable NaOH. - Calculation, ] Data (this work).
Figure 9. Silicate solubility in neutral solutions of NaNO3. - Calculation, ] Data from ref 7.
uncertainty of the measured pH but also the measured values of silicate concentrations and initial hydroxide, all of which affect the calculated values. Solubilities in solutions containing NaNO3 and NaOH are shown in Figures 9-12. The calculated values in only nitrate (Figure 9) and only hydroxide (Figure 10) closely track the data. In solutions containing both nitrate and hydroxide (Figures 11 and 12), the calculated values match the solubility data well, indicating a good fit for the regressed parameters. The use of logarithmic concentration scales in Figures 11 and 12 has both advantages and disadvantages. It certainly demonstrates that model predictions match data through nearly three decades. However, it also hides data scatter s the reality that predictions may deviate from data by 50-100% of the concentration. A glance at the last column of Table 3 indicates that a significant amount of the error could be with the data
Figure 12. Silicate solubility in solutions of 3 M NaNO3 and variable NaOH. - Calculation, ] Data (this work).
itself. For both experimental sets (1 M NaNO3 and 3 M h (average charge per Si atom in NaNO3) the value of Z the total solution) begins very small, increases to a peak that is larger than Z h ) 2, and then declines somewhat. The pattern (rise, peak, decline) is consistent with observations of many other researchers; however, the peaks here greatly exceed others’ results (which generally lie in the vicinity of Z h ) 0.75 for pH < 11.5). Hence, the uncertainty in the data may be as large as the concentration values themselves. These discrepancies illustrate the difficulties of measuring silica concentration (often due to suspended colloids and the steep solubility increase in this pH range) which have been experienced by many and contribute to high uncertainties in virtually all solubility data.
6976 Ind. Eng. Chem. Res., Vol. 42, No. 26, 2003
In contrast, the calculated values for Z h rise monotonically, never reaching a peak. They will asymptotically approach Z h ) 2 in very high caustic, consistent with experimental data. This behavior is experienced by all other models as well and appears to be unavoidable for a model that is based strictly on equilibrium thermodynamics. While it is somewhat disconcerting that the rise-fall-rise of experimental solution Z h cannot be reproduced, the resulting models are able to make surprisingly good solubility predictions throughout this transitional pH range. Conclusion Many waste processing operations require reliable predictions of silicate solubility in solutions containing high concentrations of both NaOH and NaNO3. Solubility data in 1 M and 3 M NaNO3 solutions have been presented with varying amounts of NaOH. A comprehensive model has been developed to describe silicate behavior that matches polymeric anions to NMR data. In addition, the model parameters were determined so as to also match pH data and solubilities in caustic solutions containing NaNO3. The model calculations were verified at 25 °C by comparison with all three types of data. Work is underway to extend the model to higher temperatures. Acknowledgment Oak Ridge National Laboratory is managed by UTBattelle, LLC, for the U.S. Department of Energy under contract number DE-AC05-00OR22725. Literature Cited (1) Iler, R. K. The Chemistry of Silica: Solubility, Polymerization, Colloid and Surface Properties, and Biochemistry; Wiley: New York, 1979. (2) Alexander, G. B.; Heston, W. M.; Iler, R. K. The Solubility of Amorphous Silica in Water. J. Phys. Chem. 1954, 58, 453. (3) Zarubin, D. P.; Nemkina, N. V. The Solubility of Amorphous Silica in an Alkaline Aqueous Media at a Constant Ionic Strength. Russ. J. Inorg. Chem. 1990, 35 (1), 16. (4) Svensson, I. L.; Sjo¨berg, S.; O ¨ hman, L.-O. Polysilicate Equilibria in Concentrated Sodium Silicate Solutions. J. Chem. Soc. Faraday Trans. 1 1986, 82, 3635. (5) McCormick, A. V.; Bell, A. T.; Radke, C. J. Quantitative Demonstration of Siliceous Species in Sodium Silicate Solutions by Silicon-29 NMR Spectroscopy. ZEOLITES 1987, 7, 183.
(6) Kinrade, S. D.; Swaddle, T. W. Silicon-29 NMR Studies of Aqueous Silicate Solutions. 1. Chemical Shifts and Equilibria. Inorg. Chem. 1988, 27, 4253. (7) Marshall, W. L. Amorphous Silica Solubilities sI. Behavior in Aqueous Sodium Nitrate Solutions; 25-300 °C, 0-6 Molal. Geochim. Cosmochim. Acta 1980, 44, 907. (8) Marshall, W. L.; Warakomski, J. M. Amorphous Silica Solubilities sII. Effect of Aqueous Salt Solutions at 25 °C. Geochim. Cosmochim. Acta 1980, 44, 915. (9) Eikenberg, J. On the Problem of Silica Solubility at High pH; PSI-Bericht Nr. 74, Paul Scherrer Institut, 1990. (10) Sefı´K, J.; McCormick, A. V. Thermochemistry of Aqueous Silicate Solution Precursors to Ceramics. AIChE J. 1997, 43 (11A), 2773. (11) Caullet, P.; Guth, J. L. Observed and Calculated Silicate and Aluminosilicate Oligomer Concentrations in Alkaline Aqueous Solutions. In Zeolite Synthesis; Occelli, M. L., Robson, H., Eds.; American Chemical Society: Washington, DC, 1989. (12) Grenthe, I.; Fuger, J.; Konings, R. J. M.; Lemire, R. M.; Muller, A. B.; Nguyen-Trung, C.; Wanner, H. Chemical Thermodynamics of Uranium; North-Holland, Amsterdam, Netherlands, 1992. (13) Sjo¨berg, S.; O ¨ hman, L.-O.; Ingri, N. Equilibrium and Structural Studies of Silicon (IV) and Aluminium (III) in Aqueous Solution. 11. Polysilicate Formation in Alkaline Aqueous Solution, A Combined Potentiometric and 29Si NMR Study. Acta Chem. Scand. A 1985, 39, 93. (14) Eriksson, G. Thermodynamic Studies of High-Temperature Equilibria. XII. SOLGASMIX, A Computer Program for Calculation of Equilibrium Compositions in Multiphase Systems. Chem. Scripta 1975, 8, 100. (15) Weber, C. F. Convergence of the Equilibrium Code SOLGASMIX. J. Comput. Phys. 1998, 145, 655. (16) Pitzer, K. S. Thermodynamics of Electrolytes. I. Theoretical Basis and General Equations. J. Phys. Chem. 1973, 77, 268. (17) Pitzer, K. S. Ion Interaction Approach: Theory and Data Correlation. In Activity Coefficients in Electrolyte Solutions, 2nd ed.; Pitzer, K. S., Ed.; CRC Press: Boca Raton, Florida, 1991. (18) Weber, C. F. Thermodynamic Modeling of Savannah River Waste Solutions; Draft letter report, April 4, 2001. (19) HSC Chemistry for Windows, Version 2.0, Outokumpu Research (P. O. Box 60, FIN-28101 PORI, Finland), May 31, 1994. (20) Weber, C. F.; Beahm, E. C.; Watson, J. S. Modeling Thermodynamics and Phase Equilibria for Aqueous Solutions of Trisodium Phosphate. J. Soln. Chem. 1999, 28 (11), 1207. (21) Weber, C. F.; Beahm, E. C.; Lee, D. D.; Watson, J. S. A Solubility Model for Aqueous Solutions Containing Sodium, Fluoride, and Phosphate Ions. Ind. Eng. Chem. Res. 2000, 39, 51826. (22) Weber, C. F. Calculation of Pitzer Parameters at High Ionic Strengths. Ind. Eng. Chem. Res. 2000, 39, 4422-6.
Received for review April 21, 2003 Revised manuscript received September 30, 2003 Accepted October 9, 2003 IE0303449
